Anisotropy Properties of Hexagonal Ferrimagnetic OxidesJ. Smit, F. K. Lotgering, and U. Enz Citation: Journal of Applied Physics 31, S137 (1960); doi: 10.1063/1.1984636 View online: http://dx.doi.org/10.1063/1.1984636 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/31/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On the Magnetic Anisotropy in Hexagonal Perrites AIP Conf. Proc. 34, 214 (1976); 10.1063/1.2946076 Magnetization Processes and Reversal in Ferrimagnetic Oxides with HighAnisotropy Field J. Appl. Phys. 39, 879 (1968); 10.1063/1.2163658 Domain Structure of Hexagonal Ferrimagnetic Oxides with High Anisotropy Field J. Appl. Phys. 37, 3826 (1966); 10.1063/1.1707934 Anisotropy Fields in Hexagonal Ferrimagnetic Oxides by Ferrimagnetic Resonance J. Appl. Phys. 35, 3482 (1964); 10.1063/1.1713254 Microwave Resonance in Hexagonal Ferrimagnetic Single Crystals J. Appl. Phys. 30, S175 (1959); 10.1063/1.2185873

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JOURNAL OF APPLIED PHYSICS S l; P P L E M EXT T 0 VOL. 3 1. "0. MAV, 1960

Anisotropy Anisotropy Properties of Hexagonal Ferrimagnetic Oxides

J. SMIT, F. K. LOTGERING, AND U. ENZ

PMlips Rese!trcli L!tbomtories, N. V. Philips' Gloeila.1npenfabrieken, Eindhoven-Netherlands

The effects of the incomplete alignment of all magnetic moments of a sample in an applied field on the torque curves are discussed. Examples are given of torque curves determined for cobalt-substituted hex­agonal compounds of the ferroxdure type with low anisotropy. It is shown that the characteristic details of these curves can be interpreted in terms of this incomplete alignment, both on a microscopic and an atomic scale.

1. INTRODUCTION

C· ONSIDER a ferromagnetic crystal having an uni­axial anisotropy energy K sin20 equivalent to an

anisotropy field H·4= 2K/M. In a strong external field making an angle a with the easy axis the magnetization is practically parallel to the field. If the field is in the direction perpendicular to the easy direction, exact colinearity exists for H? H·4 and the torque is zero. The torque curve as a function of a is essentially sinusoidal. If however, H < H A and a = 90°, the magnetization mnnot be aligned along H, but takes up an intermediate angle 00 given by

sinOu= H / HA. (1)

In this position, the torque is not zero but is given by

To=HM cosO(J= ±HM[1-(H/HA)2J!. (2)

Jf 0: is slightly less than 90°, 011 <90° and the torque is positive. If a is increased to a value greater than 90°, the equilibrium angle 00 changes abruptly to 180- 0o, and the torque is thus reversed. A torque curve of the type ~iven in Fig. 1 results. In crystals this sudden change will occur by wall motion and may be accompanied by hysteresis. The aim of this paper is to show that similar behavior can occur on a microscopic, and even on an a~omic scale, giving rise to peculiar anisotropy proper­ties of the material.

II. MICROSCOPIC MAGNETICALLY HARD INCLUSIONS

~et us assume that we have a magnetic crystal in which there is a cluster formation or an inclusion of a ~elatively hard ferromagnetic material with an anisot­ropy field HA. One may expect therefore that a field smaller than H_4 but larger than the bulk anisotropy ~eld will not be able to rotate all the magnetic moments 1l1to the perpendicular direction. This will not occur ~en if there is exchange coupling between the moments i.e the precipitate or cluster ions and those of the matrix.

t Us consider a spherical particle, with radius R small ,~n~ugh to assure essential parallelism of the spins lllsid . .. . e It. The particle is coupled by exchange energy '\\lIth. the surrounding matrix so that the total energy .9Utslde the particle is given by

f '" [1 (d<P)2 ] E=47r R r2 2AM2 -;;; -HM COS<p dr, (3)

in which cp(1') is the angle of the spin with the applied field. The dipolar energy is neglected. This gives for the equilibrium orientation at any point

(d2 <p 2 d<p)

AM2 -+-- -HMsin<p=O. dr2 l' dr

(4)

For <p<90° we approximate sincp by cp, and obtain then

(5)

where C is a constant and

( H)1 1( H)! q= AM =~ HE ) (6)

since AM=a2HE, where a is the atomic spacing and HE the exchange field. This configuration may be considered a three-dimensional Bloch wall. At r=R, the total surface exchange torque must be equal to the torque exerted on the particle by the crystal anisotropy defined by K=!H'~M and by the field

-47rR2AM2( dcp) dr r~I1

K

47r =-R3[K sin2(a- <p)-HM sin"'],.=R. (7)

3

T

if-,-----H V=:J 780 0

a _a V , , , , , , \M'

FIG. 1. Torque curves for an uniaxial ferromagnetic crystal. The applied field H is smaller than the anisotropy fieJd fJA.

137S

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138S SMIT, LOTGERI~G. A:\D E:\Z

FIG. 2. In the upper curve the anisotropy torque and the ex­change torque are plotted for a hard magnetic sphere in a soft magnetic matrix, whereas in the lower curve is shown the resulting torque as function of the orientation of the applied field.

If the angle that the particle magnetization makes with the easy axis is 8==a- <Pr~R, then, according to Eq. (5) and replacing the sine by its argument in the last term in Eq. (7):

Sin2e=6(~)2HE R HA

X[1+~(~)~+~(R)2~](a_8). (8) G HE 3 G HE

The left and the right hand side of Eq. (8) have been plotted as a function of e in Fig. 2 for different values of a. For small values of a, there is only one point of intersection, corresponding to one stable value of 8. For increasing values of a, there may occur three points of intersection, of which the upper one corresponds to the state with the lowest energy for a ~ 90°. For a= 90° the two extreme crossiing points are stable, but there is no easy way of switching. This has to wait until the straight line has become a tangent of the sine function. The corresponding torques as a iunction of a are plotted in the lower part of Fig. 2. For decreasing a, switching occurs for a<90°, so that hysteresis occurs. It is seen that the hysteresis is absent when there is only one crossing point at a time, if the slope of the straight line exceeds the maximum slope of the sine curve; that is, if

( G)2 [ R(H)i 1(R)2H] HA<3 - HE 1+- - +- - -_, R a HE 3 a llli

(9)

which for small fields essentially reduces to

(l()

and for strong fields to

(11) If one starts the measurements at a=90°, which' thought to be the difficult direction, then as manlS domains point up as downwards, i.e., are in the upper'Y in the lower point of intersection of Fig. 2. The torq~S curve between the switching points is then the averag ~ of the two, whereas beyond the switching points the original curves are followed. This torque curve, there~ fore, has the remarkable property that it has a positive slope at a= 90°, indicating the existence of a preferential direction. This is only true if the value of 8 at a = 90° is smaller than 45°.

This behavior may also be demonstrated in Fig. 3. Two preferential directions for the particles, pointing in arbitrary directions are indicated. \Vhen, after de­magnetization, the applied field is along the bisector the magnetizations point along MJ and M 2• Because of symmetry there is no torque. If the field is rotated through a small angle, the magnetization vectors rotate in the sense indicated, because the exchange "spring" of M 1 is released and the other is stretched. It is seen that for large anisotropies the resulting torque tends to restore the crystal axis in the original orientation with respect to H, as is the case for an easy direction of magnetization.

Such phenomena as hysteresis and the occurrence of an easy direction in an intrinsically abhorred direc­tion are observed in compounds with the hexagonal magnetoplumbite structure and the general chemical formula BaFe12_2xCoxTixOI9. It is knownl that the substitution of iron by cobalt reduces the crystalline anisotropy constant KJ, and that Kl becomes negative at low temperatures for x> ~ 1. Torque measurements

K

--- FIG. 3. Schematical repre-sentation of the orientation of

~'--------""'" H the magnetic vectors of two types of hard magnetic par· ticles in a field.

K

1 E. W. Gorter, see H. B. G. Casimir et aI., Proc. Magnetism Conference, Grenoble, France, July 1958, p. 303.

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.A:\ ISO T R 0 P Y 0 F HEX AGO 1\ A L FER R I :\1 A G NET reo X IDE S 139S

FIG. 4. Torque curves for BaFe..,Cou Tiu 0 19 in a field of 32000 oe at T=90oK for rotations in the basal plane. The samples were heated to room temperature before each series of measurement.

were carried out (Fig. 4) on polycrystalline samples with compositions near the compensation point for K 1. The c axes of the crystals of the samples were oriented by the so-called topotactical reaction.2 This method is essential for these low anisotropy materials because direct orientation of the crystals in a field is not effec­tive. Figure 4 gives the torque curve for x= 1.1 at T=900K in fields of 32000 oe.

Cobalt-rich clusters presumably play the role of the hard magnetic particles here. The anisotropy of Co is very high. (This is discussed in the next section.) When the measurements were started with the field along the c axis, this direction proved to be an easy direction for small deviations. The slope of the torque CUrve at 0=0 is, however, much larger than that of the torque curves obtained when starting with the field in the basal plane. At room temperature no anomalous properties were observed. It may be expected that con­?ition (10) then applies because the anisotropy of cobalt lS strongly temperature dependent. Apparently at T""90oK the applied field of 32000 oe is smaller than that given by (11).

It will be shown in the next section that although the resultant anisotropy of all cobalt ions is negative, each Co ion is not isotropic in the basal plane. Most of them have one easy direction near to the basal plane. The anisotropy in' the basal plane is then to a first appro xi­lllation cancelled out by the combined action of all ions. If cluster formation takes place, however, it may also he. expected that clusters occur with specific preferential otJentations in the basal plane again giving rise to ---'F. K. Lotgering, J. Inorg. Kuclear Chern. 9, 113 (1959).

hy~teresis (in lhi" case, rotational hysteresis). This was only observed in relatively small fields, in which the material was not saturated «3000 oe). At strong fields (32000 oe), however, torque curves of the type of Fig. 5 were observed; that is, a more or less normal torque curve with 1800 symmetry was found, with the direction in which the field was first applied as easy axis. Two such curves are shown, obtained at 900 K after heating the specimen to room temperature. The material always remembers the axis of the first field to which it has been exposed after demagnetization. It possesses, as it were, an absolute memory.

A tentative explanation of this behavior is the following. Let us assume that the clusters with different easy axes in the basal plane are not too far apart, so that the exchange interaction tends to align them parallel. If a strong field is applied, an equilibrium configuration will be established in which the total energy (consisting of field energy, anisotropy energy, and exchange energy) is minimum with respect to the slight rotations of the field, as has been discussed (Fig. 3). This means that the magnetization of any individual cluster is not freely rotatable. If the field is rotated and passes the direction perpendicular to the cluster, which originally made the largest angle with the field, this cluster will not now reverse its magnetization because it is fixed to the neighboring clusters. The configuration can only rotate as a whole, maintaining the mutual orientations of the clusters, when the per­pendicular orientation of an appreciable part of all clusters is passed. In this way one, obtains an anisotropy with 180° symmetry. The observed torques are of the same order of magnitude as the amplitude of the hysteresis in Fig. 4.

III. ANOMALOUS TORQUE CURVES ON AN ATOMIC SCALE

The theory of the crystalline anisotropy energy for the COlI ion in an octahedral site in the spinel lattice has been worked out by Slonczewski.3 It was shown

dyne cmlgr

Ba reS•8Co,., 7iii1S T=90oK

H=32000 Oe

Torque in Basal-plane

o

FIG. 5. Torque curve for BaFe9.8C0i.1TiJ.1019 at T =900 in the bacal plane for difierent orientations of the field which has been first applied after demagnetization.

3 J. C. Slonczweski, Phys. Rev. 110, 1341 (1958).

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140S S:VIIT, LOTGER[.:-JG, AND ENZ

K L

S ~====~H.M

(a)

'---'---H,M

Rare Earth

FIG. 6. Schematic representation of the orientation of the spip and orbital angular momentum vectors with respect to the aXIS of an uniaxial potential in an applied field for the COIl ion and an typical rare earth ion.

there that the ion in its lowest state has an orbital angular momentum, which is quantized as if it were in a P state. Due to the action of a trigonal electrostatic field, the orbital momentum is strongly bound to one of the [l11J axes of the crystal. The torque exerted by the field on the crystal is then caused by the coupling of the spin to the orbit given by XLS and by the coupling of the spin to the field. If we consider only one Co ion in a matrix, the coupling with the magnetization of the matrix (which is aligned along the field) by means of the exchange energy] is more important than the action of the field itself on the Co ion. In a series of springs the force for a certain deformation is determined by the weakest spring. In this case this is the spin-orbit energy X, which is about ten times as small as the other inter-

dyne cm/gr

i'tT ox,

-10 7

Theoretical Torque for Co in

Ba Fe'2-2x Ti~ Cox 0,9 x,+6xz=x

T=90 oK /0:)./ =132cm-f

FIG. 7. Theoretical torque curves at T=90oK for Co ions in different sites in hexagonal compounds.

a~tions. Schematica.lly o.ne thcn obtain.s the picture.~ fig. 6(a). In passmg It may be said that SillliIll b~havior is expected for s~me rare earth io~s with t!!l difference that then ] IS the smallest mteract~ [Fig. 6(b)].~~j

I~ may ?e assumed that t~e orb~tal rr:om~nt of tl CO-Ion SWItches to the opposite onentatIOn If H, a~ therefore 5, passes the direction perpendicular to ~ trigonal axis (0=90°). The torqu.e ~as a value of a~ taA sinO (5=! and a factor a IS m~luded because1'Xl refers to the free ion) and reverses SIgn at 8"", 900 • 1; the atomic case, no hysteresis is involved because of1~ quantum mechanical tunnel effect (not discussed heret; Actually, at finite temperatures the ion is not comple~!i in its lowest state and the torque has to be multip1i~ by a factor tanh!;aAI (cosO~/kT. For T=900K'an'~ using Slonczewski's value of i aX I = 132 cm-1 the up~ curve of Fig. 7 is obtained. The characteristic feature. the curve of Fig. 1 is still present although the sha~ edges are rounded off.::~

We now apply these theoretical results to the torqJfi curves of some cobalt substituted ferroxdure samplesili! the type discussed in Section 2. For the compositi~ BaFe12-2xCoxTix019 torque curves, averaged for clocla~l wise and contour clock-wise rotation, are given iii Fig. 8 for x=0.9, 1.0, 1.1 lying in the compensati(i~ region, and for x= 1.5. The influence of the Co ions ~ the torque curves is particularly strong near 8=309'P for x= 1.0 there is both a preferential direction (c axiS){ and a preferential cone. Expansion in a Fourier seri~'; would give terms up to at least sin128 and is therefote~ not very appropriate. In order to understand tbe~~~ curves we have to consider the atomic structure of tb~; compounds.4 Like the spinels, they are composed of~ close-packed structure of oxygen ions, some of t~ interstices being occupied by the metal ions. In sonitiJ layers a fraction of the oxygen ions is replaced by" ions, rendering the structure hexagonal. The layers between have the spinel structure, with one of the [11 axes parallel to the c axis. The position of the octahe ions is indicated in Fig. 9. The trigonal axis of central ion is parallel to the c axis and those of others make angles of about 71 ° or 109° with the c The latter metal ions do not find themselves in a fie, of purely trigonal symmetry around their appropriat [111J axes because one octahedral ion is missing ~n t environment of each octahedral ion but a Ba IOn present at a greater distance. We shall neglec~ til';; difference since Ti ions and other Co ions also dlstol~ the crystalline fields. A similar situation is present ..... j~ disordered inverse spinels and there the effect of Co.;~t the anisotropy is of the same type as in magnet~ ~ though somewhat weaker. The central octahedral ~0c,~ together with the three-upper ions of Fig. 9 would ~v,~~ no uniaxial anisotropy. Therefore, for equal occupat10~~

11 4 J. J. Went, G. W. Rathenau, E. W. Gorter, and G. W. 'O, Oosterhout, Philips Tech. Rev. 13, 194 (1952).

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A N ISO T R 0 P Y 0 F HEX AGO N A L FER RIM A G NET leo X IDE S 141S

number, the influence of the noncentral ions on the uniaxial anisotropy is predominant. Since the central ions give rise to a positive K I , we can already under­stand why lhe overall effect of Cobalt on the anisotropy is negative .. It is assumed that the greater part of the Co ions goes into the spinel layers, as one should expect from electroneutrality reasons. We shall show that also the more detailed structure of the torque curves can be understood on this basis, by summing the the torque curves of the individual ions. That of the central ions is already given in the upper curve of Fig. 7. For an ion having an axis making an angle of 109° with the c axis a torque reversal will occur at 0= 19°. This is a very characteristic effect and is still found in the torque curve obtained by averaging over all azimuthal angles and taking into account the finite temperature (Fig. 7, lower curve). We have now to superimpose both curves and this is done in Fig. 9 for different ratios of the two occupation numbers for x= 1. It is satisfying that the right type of curve is found for approximately equal values of Xl and X2. Of course the positive sin20 curve of the iron ions has still to be added. The calcu­lated effect of the Co ions is about 2-3 times too large to explain the experimental results. This may be com­pared with the reduction factor 4 in nickel ferrite with respect to magnetite in which Co ions are substituted."

C-axis

105

dyne cm/9r 705

-105

T=900K H=32000 Oe 8a re72-2x Tr~Q:;PI9

705 x=7.7

-705

705

x=7.5

0

105

1'1C8T . . . orque curves for compounds BaFel2_2r COr TI,Ol!l for ,~, 1, 1.1 and 1.5 at T=90oK and H=32 000 oe.

·C. M. van der Burgt, Philips Research Repts. 12,97 (1957).

dyne cmjgr

106 ~-----------r----""""""

Calculated Torque for Cobalt in BQF~liCoo,9

T= 90 oK ~o ~o

~~~~~~------~~+-r-~~ e--

- f06 ~ __ -\II._-+-____ ..,L.j.-J-Hf---_--1

-3·1051--~~---\-+-

X2~~2 X2 , , :X

X2

_____ I Xz

X2

X1+6X2=X=!

FIG. 9. Theoretical torque curves at T=90oK for Co ions in BaFetoCoTi019 for various distributions over the octahe::lral sites in the spinel layers.

In conclusion we can say that the large negative slope near 8=90°, giving rise to a preferential cone in some cases, is due to the central octahedral ions, whereas the small or positive slope near 0=0° and the large dip around 0= 20 or 30° is due to Co ions on the other sites. On the basis of this model one should not expect a preferential cone to occur at small angles (e<200).

Finally, we will briefly consider the dynamic behavior of our simple model. If we apply a static field at right angles to the easy axis and a superimposed transverse microwave field, the orbital angular momentum of the Co ions will change sign periodically and, with it, the sign of the large torque. In the first approximation we already have this large amplitude of the torque for a very small alternating field, as if a very large opposing static field were present. This can in principle be identified with the so-called giant anisotropy fields found by Dillon6 and explained by KitteJ.7 For a purely uniaxial field the situation is as simple as treated here, and should apply to Co ions in octahedral sites in spinels or similar compounds. For the rare earth ions in garnets, the crystalline fields do not have this simple symmetry. Consequently more orbital states are stable, and switching at intermediate angles may occur.

6 J. F. Dillon, Phys. Rev. 105, 759 (1957). 7 C. Kittel, Phys. Rev. Leiters 3, 169 (1959);]. Appl. Phys. 31,

l1S (1960), this issue.

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